MNRAS 429, 2894–2902 (2013)
doi:10.1093/mnras/sts471
An improved model for the infrared emission from the zodiacal dust
cloud: cometary, asteroidal and interstellar dust
M. Rowan-Robinson‹ and B. May
Astrophysics Group, Blackett Laboratory, Imperial College of Science Technology and Medicine, Prince Consort Road, London SW7 2AZ
Accepted 2012 November 21. Received 2012 November 19; in original form 2012 October 5
ABSTRACT
We model the infrared emission from zodiacal dust detected by the IRAS and COBE missions,
with the aim of estimating the relative contributions of asteroidal, cometary and interstellar
dust to the zodiacal cloud. Our most important result is the detection of an isotropic component
of foreground radiation due to the interstellar dust.
The dust in the inner Solar system is known to have a fan-like distribution. If this is assumed
to extend to the orbit of Mars, we find that cometary, asteroidal and interstellar dust account for
70, 22 and 7.5 per cent of the dust in the fan. We find a worse fit if the fan is assumed to extend
to the orbit of Jupiter. Our model is broadly consistent with the analysis of interplanetary dust
detected by Ulysses and other spacecraft. Our estimate of the mass–density of interstellar dust
in the inner Solar system is consistent with estimates from Ulysses at 1.5 au, but is an order
of magnitude higher than Ulysses estimates at r > 4 au. Only 1 per cent of the zodiacal dust
arriving at the earth would be interstellar, in our model.
Our models can be further tested by ground-based kinematical studies of the zodiacal cloud,
which need to extend over a period of years to monitor solar cycle variations in interstellar
dust, by dynamical simulations, and by in situ measurements from spacecraft.
Key words: zodiacal dust – interplanetary medium – planetary systems.
1.2 Kinematical studies
1 I N T RO D U C T I O N
The zodiacal dust is an important constituent of the Sun’s debris
disc, and contains clues to the recent history of that disc. In this
paper, we attempt to estimate the relative contributions of asteroidal,
cometary and interstellar dust to the infrared emission from zodiacal
dust detected by IRAS and COBE. This is the first attempt to model
the data from both missions simultaneously.
1.1 Pre-IRAS
Early work on modelling scattered optical zodiacal light was extensively reviewed by Giese et al. (1985), Giese, Kneissel & Rittich
(1986) and Leinert (1985). The consensus from these studies was
that the number density of dust grains in the inner Solar system
followed a distribution
n(r) = n0 r −γ f (β0 ),
(1)
where r is the distance from the Sun in au and β 0 is the elevation
from the symmetry plane of the dust, with γ ∼ 1.3.
A popular form for f(β 0 ) was the ‘fan’ distribution
f (β0 ) = exp − P |sinβ0 |.
Early work on the kinematics of the zodiacal dust was begun by the
Imperial group in the late 1960s (Reay & Ring 1968; Hicks, May
& Reay 1974; East & Reay 1984). Hicks et al. (1972, 1974) raised
the possibility that some of the zodiacal dust in the vicinity of earth
may be of interstellar origin, an idea that was developed more fully
by May (2007). May’s estimate was that the kinematic asymmetry
seen by Hicks et al. (1974) between 1971 September–October and
1972 April could be explained if ∼10 per cent of the dust were
part of a linear flow, due for example to interstellar dust, with the
rest being due to dust in circular orbits, arising from cometary and
asteroidal sources.
More recent kinematic measurements of zodiacal dust have been
made by Reynolds, Madsen & Moseley (2004), using the Wisconson H-alpha Machine (WHAM) spectrometer, and these observations have been modelled by Madsen et al. (2006) using models
of cometary and asteroidal dust published by Ipatov et al. (2006).
However May (2007) still found that there was a discrepancy between the observations of Reynolds et al. (2004) and those of East &
Reay (1984), which might still leave room for a seasonal variation
in zodiacal dust kinematics due to the presence of interstellar dust.
(2)
1.3 IRAS
E-mail: [email protected]
Our understanding of the zodiacal dust was transformed by the
IRAS mission in 1983, which gave the first all-sky maps of the
C 2013 The Authors
Published by Oxford University Press on behalf of the Royal Astronomical Society
Infrared emission from the zodiacal dust cloud
zodiacal dust emission and discovered both the IRAS Zodiacal
Bands, attributed to collisions between members of asteroid families (Dermott et al. 1984; Low et al. 1984; Sykes & Greenberg 1986;
Sykes 1990; Grogan, Dermott & Xu 2001), of which at least five
are now known, contributing about 10 per cent of the total zodiacal emission in the infrared, and, on a small scale, cometary dust
trails (Sykes et al. 1986). These demonstrated that both collisions
between asteroids and comet debris must, at some level, contribute
to the zodiacal dust cloud. IRAS also revealed a local ring of dust
near the Earth (Dermott et al. 1994; Leisawitz et al. 1994; Reach
et al. 1995).
The first full models of infrared emission from the zodiacal dust
cloud were given by Rowan-Robinson et al. (1990, 1991) and Jones
& Rowan-Robinson (1993). The novel features of these models
were as follows.
(i) Use of a modified fan distribution
f (β0 ) = (cosβ0 )Q exp − P sin|β0 |ξ .
(3)
This incorporates the (cosβ 0 )Q term introduced by Murdock &
Price (1985) and the smoothing at the symmetry plane introduced
by Deul & Wostencroft (1988), where
ξ = 2 − |z/z0 |,
for
|z| < z0 , = 1 otherwise,
and z0 = 0.065 au.
Dust grains were assumed to be grey (Qν = 1) at 12–100 μm, so
their typical radii needed to be > 10μm.
(ii) Detailed models of the two strongest pairs of narrow asteroid
band features at |β = 1.◦ 4 and 10.◦ 2.
(iii) Identification of an additional component of broad asteroidal bands centred at |β| ∼ 10◦ . Rowan-Robinson et al. (1991)
suggested that the cooler spectral energy distribution of the broadbands pointed to a distant origin for these bands, perhaps at ∼50 au,
but a parallax test performed by Jones & Rowan-Robinson (1993)
using the third IRAS HCON showed that they are consistent with
being, like the narrow bands, at r ∼ 1.5–3 au. Jones & RowanRobinson’s (1993) fan model extended to r = 1.5 au and they found
that asteroidal dust could account for ∼25 per cent of the dust involved in the fan component. So a major deficiency of that model
is to account for the remaining ∼75 per cent of the dust in the fan.
Liou, Dermott & Xu (1995) used a dynamical analysis fitted to
the IRAS data to conclude that 74 per cent of the zodiacal cloud
was of cometary origin, while 26 per cent was asteroidal. Durda
& Dermott (1997) concluded that as much as 34 per cent could be
asteroidal.
Nesvorny et al. (2010) have modelled the dynamics of cometary
dust and compared results with IRAS data, concluding that over
90 per cent of zodiacal dust is cometary. However, their use of
smoothed data means that the narrow asteroidal bands are smoothed
out and it is likely that they have underestimated the contribution of
asteroidal dust.
1.4 Ulysses
The next major contribution to understanding of the nature and origin of zodiacal dust was in situ measurements of zodiacal dust by
Ulysses (1990–2009) and other spacecraft. Divine (1993) analysed
Ulysses and other data to identify five different components of interstellar dust. The three dominant components are the core (fan)
component, extending to ∼2 au, an asteroidal component dominating from 1–3 au and a relatively isotropic halo component which
2895
becomes significant at ∼3 au and dominates at large distances (see
also the review of Solar system dust by Grun et al. 1993).
One of the major discoveries of Ulysses was the strong role
of interstellar dust in the Solar system, especially at greater distances from the Sun (see the review by Mann 2010). Interstellar
dust has also been detected in the Solar system by the Galileo,
Cassini and Stardust spacecraft (Altobelli et al. 2003; Grun et al.
2005; Kruger et al. 2010; Mann 2010; Sterken et al. 2012a). The
density of interstellar dust in the Solar system varies with time,
with a strong dependence on the solar cycle. Because the dust particles become charged, their motion is strongly influenced by the
plasma flow, and the Earth’s bow shock excludes much of the dust,
especially the smaller particles. At times of solar maximum, magnetic field reversals mean that interstellar dust is allowed into the
inner Solar system. Its relatively high velocity relative to the Sun,
typically ∼26 km s−1 , means that this dust can traverse the ∼200
au from the magnetopause in ∼50 yr. By contrast dust in approximately circular orbit around the sun, and spiralling inwards under
the influence of the Poynting–Robertson effect, takes ∼10 000 yr
to travel from the asteroid belt to the Earth (Gustafsen, Misconi &
Rusk 1987).
Interstellar dust within a cylindrical column with the Hoyle–
2
∼ 4 au (Hoyle & Lyttleton 1939)
Lyttleton radius 2GM /v
would be gravitationally trapped by the Sun and can in principle be
an important supply source for the zodiacal dust cloud. Since the
time-scale for this gravitational trapping is of the order of ∼2 yr, the
quantity of interstellar dust in the inner Solar system may vary on
a time-scale of years, and would certainly be expected to vary over
the solar cycle. Interstellar dust with impact parameter appreciably
greater than 4 au would pass through the inner Solar system rapidly
but may still contribute to the infrared emission detected by IRAS
and Diffuse Infrared Background Experiment (DIRBE).
The planets have a negligible effect on the motion of the interstellar dust. By contrast the very slow passage of dust grains spiralling inwards under the Poynting–Robertson effect ensures that
each planet can significantly modify the orbits of dust grains and
the vertical distribution of dust density.
Grogan, Dermott & Gustafson (1996) simulated the flow of interstellar dust into the inner Solar system, including the effects of
gravity, radiation pressure and magnetic forces on the dust grains.
They found that the distribution of this dust tends to be approximately isotropic over most of the sky, with a column downstream
of the flow direction in which larger grains (0.3 μm) are focused
by the Hoyle–Lyttleton accretion, and from which smaller grains
(<0.3 μm) are excluded by the combined effects of radiation pressure and magnetic forces. They concluded that the contribution of
local interstellar grains to the infrared foreground would be small
(<0.1 MJy sr−1 at 12 μm), but this was based on rather a low estimate of the density of interstellar dust at 5 au (see Section 6 below).
A more detailed simulation of the flow of interstellar dust through
the Solar system, under the action of gravity, radiation pressure and
Lorentz forces, through the different phases of the solar cycle, has
been given by Sterken et al. (2012b). They find strong variation
with time of the density of interstellar dust in the inner Solar system. They also find some systematic deviation from isotropy with
time during the solar cycle, but this is unlikely to be detectable in
the type of modelling we are carrying out here.
1.5 COBE
The DIRBE instrument on COBE mapped the zodiacal emission in
1989–90 with a different scan strategy to IRAS. Whereas IRAS
2896
M. Rowan-Robinson and B. May
scanned from ecliptic pole to pole, the COBE spacecraft executed an additional circular motion which resulted in sampling the
zodiacal emission at a wide range of solar elongations every day.
A further advantage was that DIRBE, unlike IRAS, had an absolute calibration. The IRAS zodiacal measurements had a somewhat
better resolution than DIRBE, and coupled with the fact that scans
were at approximately constant solar elongation, this meant that
IRAS gives better resolution of the fine structure in the zodiacal
bands.
Kelsall et al. (1998) modelled the zodiacal emission detected by
DIRBE using a slightly different modified exponential fan to Jones
& Rowan-Robinson (JRR) with
f (β0 ) = exp −P (sin |β0 | − z0 /2)γ1 ,
γ1
= exp −P (sin |β0 |/2z0 ) ,
2
for
for
sin |β0 | < z0 ,
sin |β0 | > z0 ,
where z0 = 0.189, γ 1 = 0.942.
No attempt was made to account for cometary or interstellar dust,
but the fan is assumed to extend to 5.2 au. Their main innovation is
inclusion of the ring of dust at r ∼ 1 au and the blob trailing behind
Earth, which had been first detected by Dermott et al. (1994) in
the IRAS data, and confirmed by Reach et al. (1995). They justified
the neglect of interstellar dust based on the work of Grogan et al.
(1996).
In this paper, we revisit models for the combined IRAS and
DIRBE observations of zodiacal dust, with a view to trying to
determine the relative contributions of asteroidal, cometary and interstellar dust. The IRAS Zodiacal History File remains a key data set
if full sampling of the asteroidal component is required. The DIRBE
data, with its absolute calibration, are essential if an isotropic component like interstellar dust is to be detected. The components in our
model are physically motivated, occupy distinct regions of the Solar system, and have different radial dependences. While dynamical
simulations are essential for understanding the evolution of zodiacal
dust, our approach provides a valid independent perspective.
2 INGREDIENTS OF THE MODEL
We aim to be strongly guided by the evidence from direct detection
of interplanetary dust. We can expect major transformation of the
vertical dust profile as each planetary orbit is crossed and so for our
purposes we must allow for significant changes at the orbit of Mars,
r = 1.53 au, and at the orbit of Jupiter, r = 5.2 au.
The main ingredients of the JRR (1993) model, the modified fan
and asteroid bands, correspond well to two of the main components
detected in spacecraft data. Their assumption that the fan terminates
at r = 1.5 au is consistent with the Divine (1993) analysis, though
we will explore here the possibility that it extends further. The radial
dependence for the fan assumed by JRR was r−1.0 , guided by the
expected behaviour of grains subject to the Poynting–Robertson
effect, but the evidence for a radial dependence of r−1.3 within 1.5
au now seems very strong and we use that dependence here. We
initially use the JRR parameters for the asteroid bands, but will
consider the possibility that the amplitudes need modification, to
take account of the other ingredients used here. We will also look
briefly at the possibility that the broad-bands extend to a greater
distance than 3.1 au.
The first ingredient not incorporated by JRR, but included by
Kelsall et al. (1998), is the solar ring and trailing blob and we have
used the formalism of Kelsall et al. (1998) for these, but allowing
the amplitudes to be a free parameter.
Secondly, we want to include a component corresponding to
cometary dust. The dominant contribution is expected to be from
Jupiter-family comets, which tend to have inclinations <30◦ , have
aphelia ranging from 5 to 30 au, and have their origin in the Kuiper
belt. For the cometary contribution to the asteroidal dust we therefore expect a flattened distribution, but perhaps not so strongly
flattened as the inner fan. We assume an exponential fan extending
from r = 1.5 to 30 au, with exponent PCOM (to be determined), and
assuming an r−1 radial distribution. Nesvorny et al. (2010) estimate
that the contribution of Oort cloud comets is negligible and we were
also not able to detect such a component.
Finally, for interstellar dust we assume an isotropic distribution,
with uniform density, extending from r = 0 to 30 au. In reality the
distribution of interstellar dust will be more complex, especially for
larger and smaller grains (Sterken et al. 2012b), but isotropy is a reasonable first-order assumption in this attempt to detect interstellar
dust through its infrared emission.
We assume that the cometary and interstellar components together contribute the ‘halo’ population identified by Divine (1993).
The crucial distinction between the components in the model lies
in their radial distribution, with asteroidal dust originating between
1.5 and 3.1 au, cometary dust extending from 1.5 au out to large
distances, and interstellar dust extending through the whole Solar
system. The fan is assumed to be supplied by the cometary and
asteroidal components and extends to 1.5 au. So the components
are quite distinct in their contribution to the infrared emission.
3 F I T S T O IRAS DATA
The data we have used are version 3.0 of the IRAS Zodiacal History
File, which lists position on the sky, UTCS and fluxes averaged
over 0.5 degree2 in the four IRAS bands. From this information the
solar longitude can be calculated. Data heavily affected by cirrus
are excluded, using a standard mask. The fan is assumed to have a
symmetry plane specified by (, i). We have not corrected for the
relatively small effect of the Sun’s displacement from the symmetry
plane of the zodiacal dust cloud. The density of the dust in the fan
is specified by equations (1) and (3). The dust grains are assumed
to be large and grey, with a temperature dependence T1 r−0.5 , with
T1 = 255 K. Variation of the temperature at 1 au, T1 , was explored
but not found to improve the fits. The fan was assumed to extend
out to a radius RMAX, initially taken to be 1.53 au.
The density of dust in the narrow bands was assumed to satisfy
equation (1), with γ = 1.0 and
fnb (β0 ) = expG(|β0 | − βnb ), |β0 | < βnb .
(4)
β b was taken to be 1.◦ 42 for the inner bands, corresponding to the
Themis asteroid family. For the outer bands, JRR assumed β b =
10.◦ 14, corresponding to the Eos family, but Grogan et al. (2001)
have shown that there is better agreement with the Veritas family,
with β b = 9.◦ 35, and we confirm here that this gives a better fit to
the IRAS data. The bands are assumed to extend from r = 1.53 to
3.1 au.
For the broad-bands identified by Rowan-Robinson et al. (1991)
and modelled by JRR, we assume that they have their own symmetry
plane characterized by (bb , ibb ). Their density is also assumed to
be of the form of equation (1), with γ = 1.0 and
2
fbb (β1 ) = exp − (β1 − βbb )2 /2σbb
2
,
(5)
+ exp − (β1 + βbb )2 /2σbb
Infrared emission from the zodiacal dust cloud
where βbb is the latitude of the peaks above the symmetry plane and
σbb is the peak width. These bands are assumed to extend from r =
1.53 au to R2BF, where R2BF is normally taken to be 3.1 au, but
the possibility of extension to greater distance is considered below.
The grains in the broad-bands were also assumed to be large and
we did not confirm the suggestion of JRR that the fits are improved
if the broad-band grains are assumed to be small.
For cometary dust we assume a density dependence of the form
equations (1) and (2), with γ = 1.0. We are assuming that cometary
dust has a flattened distribution, but not as strongly flattened as the
main zodiacal fan. Cometary dust is assumed to extend from r =
1.53 to 30 au.
For interstellar dust we assume an isotropic and uniform distribution, extending from r = 0 to 30 au, based on the simulations
of Grogan et al. (1996). We discuss below the expected departure
from isotropy in a cone downstream of the Sun. The outer cutoff
for cometary and interstellar dust, corresponding to the inner edge
of the Kuiper belt, is arbitrary, and the fit to the IRAS data is not
sensitive to this parameter. The Grogan et al. (1996) estimate of
a negligible contribution by interstellar dust to the infrared foreground was based on a rather low density of interstellar dust, 2 ×
10−27 gm cm−3 , at 5 au. Kimura, Mann & Jessberger (2003) and
Mann (2010) report estimates 2–20 times higher than this and we
believe it is worth allowing the density of interstellar dust to be a
free parameter and see whether the resulting implied densities are
unreasonable.
The interstellar dust grains can be expected to be smaller than
those of cometary and asteroidal origin, and are indeed found to
be so in the spacecraft studies (e.g. Grun et al. 1993). Our best
solution was found to be with Qν ∝ ν for λ > 24 μm, = 1 for λ ≤
24 μm, corresponding to grain radius ∼2–4 μm. The corresponding
temperature distribution for these smaller grains was taken to be
T = 305 r−0.4 K.
Finally, the ∼1 au ring and trailing blob are modelled using the
formalism of Kelsall et al. (1998), but we allow the amplitudes to
be a free parameter.
We estimated the rms fluctuation in each band at |b| > 40◦ , |β| >
20◦ (after subtraction of the fan contribution) to be σ ν = 0.42, 0.55,
0.45 and 1.10 MJy sr−1 at 12, 25, 60 and
100 μm, respectively.
We then minimize χ 2 = [Iν,I RAS − Cν n(r, λ, β)Bν (T (r))ds +
Dν ]2 /σν2 . Because IRAS did not have an absolute calibration there
is uncertainty in the zero-point in each band, with a quoted rms
0.66, 0.73, 0.38 and 1.19 MJy sr−1 at 12, 25, 60 and 100 μm. This
makes the estimation of any isotropic component like our assumed
interstellar component problematical.
The approach we have followed here, is to use the offsets Dν
estimated for the IRAS data by the DIRBE team from direct comparison of DIRBE and IRAS data. These are given as −0.48, −1.32,
0.13 and −1.47 MJy sr−1 at 12, 25, 60 and 100 μm (Beichman
& Wheelock 1993, IRAS project note, www.ipac.caltech.edu/ipac/
newsletters/oct93/cobe.html). We then minimize the total χ 2 ,
2897
summed over the four IRAS bands, allowing a free calibration factor
Cν .
Any isotropic component has to be consistent with the limits on
an isotropic component set by DIRBE, which are given as (2σ ) 1.9,
4.2, 1.5 and 1.3 MJy sr−1 at 12, 25, 60 and 100 μm (Hauser et al.
1998) (see Section 5).
We find that both cometary and interstellar dust components were
positively detected, in that the fit improved significantly when they
were included. This is consistent with the results of direct detection
of interstellar dust by Ulysses and other spacecraft (Grun et al.
1993).
4 R E S U LT S O F F I T S T O IRAS DATA A L O N E
The results of varying the parameters of the model are shown in
Table 1, with the penultimate column showing the sum of the reduced χ 2 summed over the four IRAS bands. The first line corresponds to the preferred model of JRR (1993), with their parameters
for the narrow and broad-bands. The second line shows the effect
of changing the radial density index to γ = 1.3.
We now add in the ring and trailing blob, cometary and interstellar
components, tune the amplitudes of these, and then solve again for
the parameters of the fan, Q and P. The fit to the IRAS data is significantly improved. We then make a number of small adjustments to the
model which improve the fit: (i) we assumed that the outer narrow
bands are due to the Veritas family, with β nb = 9.◦ 35 and adjusted
G2 to 0.12; (ii) we tuned the parameters for the symmetry plane
of the broad-bands and found a better fit at bb = 110◦ , ibb = 2.◦ 6;
(iv) we tuned the parameters for the symmetry plane of the fan and
found = 78◦ , i = 1.◦ 50. A multiparameter grid-search fitted to the
IRAS data with parameters P, Q, z0, AMPBB, COM1, for different
values of ISD1, yielded uncertainties of 2 per cent for P, Q, and 10
per cent for other parameters. We estimate the uncertainty in ISD1
as 20 per cent. The best-fitting parameters are shown in line 3 of
Table 1, shown in bold, model A.
Fig. 1 shows latitude profiles in the four IRAS bands for a scan
with solar longitude 90.◦ 61, compared with the predictions of model
A. The fit is excellent. Figs 2 and 3 show latitude profiles at 25 μm
for scans spread throughout the survey, after subtraction of the fan
component, compared with the predictions of the other ingredients
of the model. The fits are spectacularly good. The adjustments to
the parameters of the narrow (asteroidal) and broad-bands, and the
inclusion of components corresponding to cometary and interstellar
dust, all contribute to the improvement of the fits compared with the
work of JRR. We found the amplitude of the trailing blob needed to
be smaller than that assumed by Kelsall et al. (1998).
For our basic model A, with the fan extending to 1.53 au, the
asteroidal dust in the narrow and broad-bands, cometary dust and
interstellar dust, integrated over all ecliptic latitudes, contribute
22.2, 70.4 and 7.5 per cent, respectively, of the density of dust,
relative to the fan. The total contribution from the three components
adds to exactly 100 per cent of the density in the fan (this was not
Table 1. Parameters for new zodiacal dust models.
Fan
Q
P
z0
RMAX
Narrow
AMP1
Bands
AMP2
Broad b.
AMPBB
Ring
AMPSR
Blob
AMPBL
Comet
COM1
PCOM
I.S.D.
ISD1
χn2
IRAS
DIRBE
8.0
8.0
10.7
2.85
2.85
2.13
0.065
0.065
0.06
1.52
1.52
1.52
0.030
0.030
0.032
0.039
0.039
0.040
0.029
0.029
0.051
–
–
0.16
–
–
0.065
–
–
0.37
–
–
2.5
–
–
0.010
5.03
4.32
1.33
(γ = 1.0)
(γ = 1.3)
1.30 A
3.5
2.5
0.15
5.2
0.035
0.035
0.030
0.17
0.06
0.0
–
0.003
2.48
1.33 B
0.0
4.14
0.19
5.2
0.035
0.027
0.0
0.17
0.16
0.0
–
0.0
3.40
1.31 K
2898
M. Rowan-Robinson and B. May
forced on the solution). However, it is unclear how much of the
interstellar dust within the Hoyle–Lyttleton column would make it
into the fan. Much could simply be channelled on to the reverse side
of the Sun. Note that at the ecliptic plane in model A, the interstellar
dust contributes only 1 per cent of the zodiacal dust density at 1 au.
5 F I T S T O D I R B E DATA
Figure 1. Comparison of IRAS scans at 12, 25, 60 and 100 µm with zodiacal
dust model A. Scan is centred at solar longitude 90.◦ 61.
We used the DIRBE calibrated data files, which are given for each
day of the mission. We first modelled the data for individual days
using our best model A from table 2, excluding days where rms2 >
5(MJy sr−1 )2 , which can arise for example because the Moon is in
the field of view during the day. Data were used only if there were
detections in all four bands at 12, 25, 60 and 100 μm, and only sky at
|b| > 40◦ was used in the zodiacal modelling. We focused on three
blocks of data where there were a significant number of contiguous
days with good data, at day numbers 19–33, 51–95 and 216–264
in 1990. 40 of the 109 d in these three blocks were excluded. For
a further 54 d analysed, distributed at random through the mission,
none satisfied our constraint. The 69 d of data used in our solution
covered 92 per cent of the sky, due to the particular scan strategy
of COBE. After exclusion of |b| < 40◦ and areas affected by cirrus,
we used 1.5 million observations in the solution, representing about
3 per cent of the total data. This is more than a factor of 20 times
the amount of data used by Kelsall et al. (1998) in their solution
Figure 2. Comparison of IRAS scans at 25 µm, after subtraction of fan component (model A), with remaining components. Scans centred at (L to R) solar
longitude 22.08, 33.40, 53.18 and 64.◦ 93.
Infrared emission from the zodiacal dust cloud
2899
Figure 3. Comparison of IRAS scans at 25 µm, after subtraction of fan component (model A), with remaining components. Scans centred at (L to R) solar
longitude 90.◦ 61. 93.38, 121.◦ 61 and 124.◦ 99.
(0.13 per cent of the total data). We made no restriction on solar
elongation angle.
We estimated the rms fluctuations for the DIRBE data at |b| >
40◦ , |β > 20◦ , after subtraction of the model A fan, as 0.80, 0.93,
0.67 and 0.80 MJy sr−1 at 12, 25, 60, 100 μm, and calculated χ 2 as
above (last column in Table 1).
Fig. 4 shows fits with models A to data at solar elongation
90±1.◦ 0, |b| > 40◦ , on 1990 Jan 19. The nature of the COBE scan
strategy do not allow pole-to-pole scans to be plotted, as for IRAS.
We have shown fits to the 4.9 μm data, but have not used the latter
in the parameter fits. The data at 140 and 240 μm were too noisy to
plot, or to use in the solution. The model fits to the COBE data are
excellent, and by excluding directions at |b| < 40◦ from the plots,
we achieve much more compelling comparisons of the models with
the DIRBE data than shown by Kelsall et al. (1998) (their fig 8).
The combination of the larger beam of DIRBE, the precessing scan
strategy of COBE, and the poorer signal-to-noise ratio compared
with the IRAS scans, means that details of the asteroidal dust bands
are much more poorly resolved by DIRBE. We were also unable
to detect the trailing blob in the DIRBE data so the amplitude of
this component could be determined only from IRAS data. Fig. 5
shows the corresponding plots for solar elongation 70 and 110 ±
1.◦ 0, illustrating that the model works well over a wide range of solar
elongation.
We should check that the magnitude of our claimed isotropic
interstellar dust component is not inconsistent with limits set in
earlier studies. The magnitude of the isotropic component due to
interstellar dust at 12, 25, 60 and 100 μm, respectively, are for model
(A) 1.59, 2.90, 0.68, 0.23 MJy sr−1 . These are consistent with the
limits on an isotropic background set by Hauser et al. (1998). Fig. 6,
which shows χn2 as a function of the amplitude in the interstellar
dust component, keeping the other parameters of model A fixed,
illustrates the strong detection of the interstellar dust component in
both IRAS and DIRBE data.
For model A, we investigated whether extending the broad-bands
to 30 au (as proposed by Rowan-Robinson et al. 1991) affected the
fit. We found no significant improvement in the χn2 for either the
IRAS and DIRBE data. We also investigated whether making the
broad-band dust grains smaller improved the fit, but again there was
no improvement. However, the origin of this broad-band component
does merit further study. It may arise from an older collision event
between asteroid family members in the main belt. A much more
distant origin in the Kuiper belt cannot be ruled out.
6 M O D E L S W I T H FA N E X T E N D E D T O 5 . 2 au
Although the direct evidence from spacecraft data is consistent with
the fan extending to r = 1.53 au, we have explored the possibility
2900
M. Rowan-Robinson and B. May
Figure 6. χn2 versus the interstellar dust amplitude, ISD1, for IRAS and
DIRBE data.
Figure 4. Comparison of DIRBE data at solar elongation 90 ± 1.◦ 0, from
2009 January 19, restricted to |b| > 40◦ , at 4.9, 12, 25, 60 and 100 µm, with
model A.
that it could extend to the orbit of Jupiter at r = 5.2 au. We assume
that 15 per cent of the zodiacal cloud interior to 1.53 au is supplied
by the asteroidal bands, so that the amplitude of the extension from
r = 1.53 to 5.2 au is at an amplitude 0.85 times that interior to 1.53
au. The main effect of such an extension is to increase the intensity
at |β| < 30◦ . For our modified fan (equation 3) to remain consistent
with observations it is necessary to change the parameters P, Q, but
also to increase z0 . Line 5 of the table shows a fit with Q = 3.5, P =
2.5 and z0 = 0.15 (model B, see Fig. 8). It was also necessary to
reduce the amplitude of the outer narrow bands, AMP2 and of the
broad-bands, AMPBB. The best combined fit to IRAS and DIRBE
data has no cometary dust beyond 5.2 au, and the amplitude of
interstellar dust is lower by a factor of 3 than in model A. The fits
to the IRAS data are significantly worse than model A. Fig. 7 (L)
Figure 7. Dust density versus radius (au) for fan, cometary, asteroidal and
interstellar dust in model A.
Figure 5. Comparison of DIRBE data at solar elongation 70 ± 1.◦ 0 (L) and 110±1.◦ 0 (R), from 2009 January 19, restricted to |b| > 40◦ , at 4.9, 12, 25, 60 and
100 µm, with model A.
Infrared emission from the zodiacal dust cloud
2901
Figure 8. Comparison of IRAS scans at 25 µm, (L) after subtraction of fan component, extended to 5.2 au (model B), with remaining components; (B) after
subtraction of Kelsall et al. (1998) fan (model K). Scans centred at solar longitude 90.◦ 61.
shows the fit to the IRAS data at 25 μm after subtraction of the fan
component.
We have also fitted the IRAS data with the Kelsall et al. (1998)
fan model, plus ring and trailing blob. For the narrow bands we
have simply used equation (4), with adjustments to the amplitudes
and to G2 (=1.0). The Kelsall et al. fan changes shape at β 0 ∼ 10◦
and thereby dispenses with the need for the broad-bands. The fit is
noticeably worse for the IRAS data, with χn2 = 3.40 (Dν = 0). The
extension of the fan to 5.2 au is not supported by the Divine (1993)
analysis.
7 M A S S – D E N S I T Y I N Z O D I AC A L D U S T
The parameter Cν can be interpreted as πa 2 Qν n0 a0 , where a, Qν
are the characteristic grain radius and absorption efficiency, n0 is
the grain number-density and a0 = 1 au. Hence we can derive an
order-of magnitude mass–density in grains, ρgr = n0 (4πa 3 ρ0 /3),
where ρ 0 is the mean density of the grain, taken to be 2.5 gm cm−3 .
For model A, in the ecliptic plane at r = 1.53 au, and using Cν at
λ = 12 μm, we find
ρgr (Qν /a(μm)) ∼ 10−23.27 gm cm−3 , for the fan
∼10−25.27 gm cm−3 , for the interstellar dust
∼10−23.67 gm cm−3 , for the cometary dust.
For comparison Kimura et al. (2003) estimate the mass–density
of interstellar dust at r = 1.5 au as 10−25.3 ± 0.7 gm cm−3 , so our
value is consistent with this for a ∼2−4μm, Qν ∼ 1. On the other
hand they estimate the mass–density of interstellar dust at r > 4 au
as 10−26.43(+0.26, −0.60) gm cm−3 , attributing the difference to gravitational focusing of the dust in the inner Solar system. Our assumed
isotropic uniform dust density is therefore an order of magnitude
higher than that measured by Ulysses at r > 4 au. Further work will
be needed to assess whether the density of interstellar grains has
a strong dependence on radial distance. Our dust density estimates
are approximate, but the satellite estimates are also indirect, based
on the measured charge.
8 DISCUSSION
Our model provides fits to the IRAS and DIRBE data significantly
better than those achieved in the previous analyses and shows good
consistency with the major components identified by Divine (1993)
in the Ulysses spacecraft data. The major new constituents compared to JRR are the interstellar and cometary components, corresponding to the ‘halo’ component of Divine (1993).
In model A, the relative contributions of cometary, interstellar
and asteroidal dust to the density of the fan at 1.5 au are 70.4, 7.5
and 22.2 per cent, and the density of the fan is exactly accounted for
by these components. This is the first analytical fit to the zodiacal
infrared data in which the origin of the fan is fully accounted for. Our
assumed isotropic uniform density of interstellar grains is consistent
with values measured by Ulysses at 1.5 au, but an order of magnitude
higher than values measured at r > 4 au. Further work will be
needed to assess whether the density of interstellar grains has a
strong dependence on radial distance, as proposed by Kimaru et al.
(2003).
Our approach gives a valid independent perspective on dust in the
Solar system to that given by dynamical simulation of the different
components (e.g. Gustafsen et al. 1987; Durda et al. 1997; Dikarev
et al. 2005; Ipatov et al. 2008; Nesvorny et al. 2010). Although
predictions of dynamical models with just asteroidal and cometary
dust have been compared with IRAS (Nesvorny et al. 2010) and
COBE (Dikarev et al. 2005) data, we find that to fit both sets of data
to the accuracy achieved here requires the additional ingredient of
a homogenous, isotropic component, corresponding to interstellar
dust.
While cometary and asteroidal dust spiral inwards very slowly
and can experience strong orbital changes at the crossing of planetary orbits, the fast-moving interstellar dust is almost unaffected
by the planets and is trapped in the inner Solar system by the sun’s
gravity through Hoyle–Lyttleton accretion. The smallest grains will
also be repelled from a cylindrical column behind the Sun by radiation pressure and magnetic forces. This column would also show
a concentration of larger grains being accreted towards the Sun.
Unfortunately the direction of this column [(λ, β) ∼ (−172◦ , −5◦ );
Kimaru et al. (2003)] is rather close to the ecliptic plane, so it
cannot be resolved in maps of the IRAS background emission. Detection of the downstream accretion column would be an important
confirmation of the interstellar dust component.
Nesvorny et al. (2010) carried out an interesting simulation of the
dynamics of cometary dust in the Solar system. Their estimate that
cometary dust provides over 90 per cent of the zodiacal dust in the
2902
M. Rowan-Robinson and B. May
inner Solar system is higher than our model predicts. We estimate
that cometary dust could contribute 60–80 per cent, with asteroidal
and interstellar dust contributing 20–40 per cent between them. Our
estimate of the relative proportions of cometary and asteroidal dust
agree well with the estimate of Liou et al. (1995) from a dynamical
model.
We find that the contribution of interstellar dust may be more
significant than estimated by Grogan et al. (1996). The DIRBE
data iare crucial here in demonstrating the presence of an isotropic
foreground component at 12, 25, 60 and 100 μm. The amplitude
of this foreground does not conflict with the isotropic background
limits set by Hauser et al. (1998).
We find strong support for the Hicks et al. (1974) proposal, extended by May (2007), based on kinematical observations, that
interstellar dust is a significant contributor to the local zodiacal
dust cloud, although the density we find in the ecliptic plane is
much lower than that estimated by May. There is a need for a new
ground-based kinematic study of zodiacal dust, ideally extended
over several years to test for time variation through the solar cycle. The models presented here can also be tested by dynamical
simulations and by further in situ measurements from spacecraft at
r > 5 au.
The zodiacal dust cloud appears to be supplied by a combination
of cometary dust, dust from collisions between members of asteroid
families in the asteroid belt, and interstellar dust. It therefore carries
detailed information about the recent history of the Sun’s debris
disc and merits far more intensive study than it has received to date.
REFERENCES
Altobelli N., Kempf S., Landgraf M., Srama R., Dikarev V., Kruger H.,
Moragas-Klostermeyer G., Grun E., 2003, J. Geophys. Res., 108, 8032
Dermott S. F., Nicholson P. D., Burns J. A., Houck J. R., 1984, Nat, 312,
505
Dermott S. F., Jayaraman S., Xu X. L., Gustafson B. A. S., Liou J. C., 1994,
Nat, 369, 719
Deul E. R., Wostencroft R. D., 1988, A&A, 196, 277
Dikarev V., Grun E., Baggaley J., Galligan D., Landgraf M., Jehn R., 2005,
Adv. Space Res., 35, 1282
Divine N., 1993, J. Geophys. Res., 98, 17029
Durda D. D., Dermott S. F., 1997, Icarus, 130, 140
East I. R., Reay N. K., 1984, A&A, 139, 512
Giese R. H., Kinateder G., Kneissel B., Rittich U., 1985, in Giese R. H.,
Lamy P., eds, Properties and Interactions of Interplanetary Dust. Reidel,
Dordrecht, p. 225
Giese R. H., Kneissel B., Rittich U., 1986, Icarus, 68, 395
Grogan K., Dermott S. F., Gustafson B. A. S., 1996, ApJ, 472, 812
Grogan K., Dermott S. F., Xu Y. L., 2001, Icarus, 152, 251
Grun E. et al., 1993, Nat, 362, 428
Grun E., Srama A., Kruger H., Kempf S., Dikarev V., Helfert S., MoragasKlostermeyer G., 2005, Icarus, 174, 1
Gustafsen B. A. S., Misconi N. Y., Rusk E. T., 1987, Icarus, 72, 568
Hauser M. et al., 1998, ApJ, 508, 25
Hicks T. R., May B. H., Reay N. K., 1972, Nat, 240, 401
Hicks T. R., May B. H., Reay N. K., 1974, MNRAS, 166, 439
Hoyle F., Lyttleton R. A., 1939, Proc. Cambridge Philos. Soc., 35, 405
Ipatov S. I., Kutyrev A. S., Madsen G. J., Mather J. C., Moseley S. H.,
Reynolds R. J., 2006, 37th Annual Lunar and Planetary Science Conference, League City, Texas
Ipatov S. I., Kutyrev A. S., Madsen G. J., Mather J. G., Moseley S. H.,
Reynolds R. J., 2008, Icarus, 194, 769
Jones M. H., Rowan-Robinson M., 1993, MNRAS, 264, 237
Kelsall T. et al., 1998, ApJ, 508, 44
Kimura H., Mann I., Jessberger E. K., 2003, ApJ, 582, 846
Kruger H. et al., 2010, Planet. Space Sci., 58, 951
Leinert C., 1985, in Giese R. H., Lamy P., eds, Properties and Interactions
of Interplanetary Dust. Reidel, Dordrecht, p. 369
Leisawitz D. et al., 1994, BAAS, 26, 1409
Liou J. C., Dermott S. F., Xu Y. L., 1995, Planet. Space Sci., 43, 717
Low F. J. et al., 1984, ApJL, 278, L19
Madsen G. J., Reynolds R. J., Ipatov S. I., Kutryev A. S., Mather J. C.,
Moseley S. H., 2006, in Dust in Planetary Systems. Kaua’i, Hawaii, LPI
Contribution No. 1280, p. 111
Mann I., 2010, ARA&A, 48, 173
May B. H., 2007, PhD thesis, Imperial College
Murdock T. L., Price S. D., 1985, AJ, 90, 375
Nesvorny D., Jenniskens P., Levison H. F., Bottke W. F., Vokrouhlicky D.,
Gounelle M., 2010, ApJ, 713, 816
Reach W. T. et al., 1995, Nat, 374, 521
Reay N. K., Ring J., 1968, Nat, 219, 710
Reynolds R. J., Madsen G. J., Moseley S. H., 2004, ApJ, 612, 1206
Rowan-Robinson M., Hughes J., Vedi K., Walker D. W., 1990, MNRAS,
246, 273
Rowan-Robinson M., Hughes J., Jones M., Leech K., Vedi K., Walker D.
W., 1991, MNRAS, 249, 729
Sterken V. J., Westphal A. J., Altobelli N., Postberg F., Srama R., Grun
E., 2012a, 43rd Lunar and Planetary Science Conference, League City,
Texas
Sterken V. J., Altobelli N., Kempf S., Schwehm G., Srama R., Grun E.,
2012b, A&A, 538, A102
Sykes M. V., 1990, Icarus, 84, 267
Sykes M. V., Greenberg R., 1986, Icarus, 65, 51
Sykes M. V., Lebofsky L. A., Hunten D. M., Low F., 1986, Sci, 232, 1115
This paper has been typeset from a TEX/LATEX file prepared by the author.